Upper bound on the number of systems of Hecke eigenvalues for Siegel modular forms (mod p)

We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result agrees with recent work of G. Herrick.Comment: 4 pages, amsart class; included reference to current work of G. Herrick; fixed a small error in the estimate of Corollary

Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms

In a letter to Tate (published in Israel J. Math. in 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) are the same as the ones given by algebraic modular forms (mod p) on a quaternionic unitary group, as defined by Gross in Israel J. Math. in 1999. The correspondence is obtained by restricting to the superspecial locus of the moduli space of abelian varieties.Comment: 28 pages; submitted to the Journal of Number Theory. Based on chapter 3 of math.NT/0306224, reworked and corrected. Comments are welcom

Distinguishing eigenforms modulo a prime ideal

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given weight and level. This was adapted by Ram Murty, Kohnen and Ghitza to the case of two eigenforms of the same level but having potentially different weights. We consider their expansions modulo a prime ideal, presenting a new bound. In the process of analysing this bound, we generalise a result of Bach and Sorenson, who provide a practical upper bound for the least prime in an arithmetic progression.Comment: 13 page

Distinguishing newforms

Let $n_0(N,k)$ be the number of initial Fourier coefficients necessary to distinguish newforms of level $N$ and even weight $k$. We produce extensive data to support our conjecture that if $N$ is a fixed squarefree positive integer and $k$ is large then $n_0(N,k)$ is the least prime that does not divide $N$.Comment: 15 pages, 8 table

The 2008 election: A preregistered replication analysis

We present an increasingly stringent set of replications of Ghitza & Gelman (2013), a multilevel regression and poststratification analysis of polls from the 2008 U.S. presidential election campaign, focusing on a set of plots showing the estimated Republican vote share for whites and for all voters, as a function of income level in each of the states. We start with a nearly-exact duplication that uses the posted code and changes only the model-fitting algorithm; we then replicate using already-analyzed data from 2004; and finally we set up preregistered replications using two surveys from 2008 that we had not previously looked at. We have already learned from our preliminary, non-preregistered replication, which has revealed a potential problem with the published analysis of Ghitza & Gelman (2013); it appears that our model may not sufficiently account for nonsampling error, and that some of the patterns presented in that earlier paper may simply reflect noise. In addition to the substantive interest in validating earlier findings about demographics, geography, and voting, the present project serves as a demonstration of preregistration in a setting where the subject matter is historical (and thus the replication data exist before the preregistration plan is written) and where the analysis is exploratory (and thus a replication cannot be simply deemed successful or unsuccessful based on the statistical significance of some particular comparison).Comment: This article is a review and preregistration plan. It will be published, along with a new Section 5 describing the results of the preregistered analysis, in Statistics and Public Polic